Universal algebra [5, 9, 4] is the theory of equalities t = u. It is a simple framework within which we can study mathematical structures, for example groups, rings, and fields. It has also been applied to study the mathematical properties of mathematical truth and computability. For example boolean algebras correspond to classical truth, heyting algebras correspond to intuitionistic truth, cyl...

This paper introduces a generalization of self-dual marked flattenings defined in the lattice of mappings. This definition provides a way to associate a self-dual operator to every mapping that decomposes an element into subelements (i.e. gives a cover). Contrary to classical flattenings whose definition relies on the complemented structure of the powerset lattices, our approach uses the pseudo...

It is known that exactly eight varieties of Heyting algebras have a modelcompletion, but no concrete axiomatisation of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in th...

In this paper, we introduce the concept of meet precontinuous posets, a generalization of meet continuous lattices to posets. The main results are: (1) A poset P is meet precontinuous iff its normal completion is a meet continuous lattice iff a certain system γ(P ) which is, in the case of complete lattices, the lattice of all Scott-closed sets is a complete Heyting algebra; (2) A poset P is pr...

In 1982, L. Iturrioz introduced symmetrical Heyting algebras of order n (or SHnalgebras). In this paper, we define and study tense SHn-algebras namely, SHnalgebras endowed with two tense operators. These algebras constitute a generalization of tense Łukasiewicz-Moisil algebras. Our main interest is the duality theory for tense SHn-algebras. In order to do this, we requiere Esakia’s duality for ...

We introduce the bimodal logic S4.Grzu, which is the extension of Bennett’s bimodal logic S4u by Grzegorczyk’s axiom ( (p→ p)→ p)→ p and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of S4.Grzu, thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logicWS5.C, which is the extens...

We show that if a subgroup of the automorphism group Fraïssé limit finite Heyting algebras has countable index, then it lies between pointwise and setwise stabilizer some set.

In their seminal paper Birkhoff and von Neumann revealed the following dilemma : “... whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic.” In this paper we eliminate this dilemma, providing a way for maintaining bo...

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper we consider the alternative approach in which all consistent sets are kept, leading to a type of ‘many world-views’ picture of t...

It is assumed that a Kripke-Joyal semantics A = 〈C,Cov,F, 〉 has been defined for a first-order language L. To transform C into a Heyting algebra C on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of C. A pretopology Cov is defined on C using the pretopology on C. A sheaf F is made up of sections of F that obey fu...